Journal of Integer Sequences, Vol. 22 (2019), Article 19.5.7

On a Family of Functions Defined Over Sums of Primes

Christian Axler
Department of Mathematics
40225 Düsseldorf


Let r and m be real numbers so that the sum Sr,m(x) = ∑px pr logm p diverges as x → ∞. Here p runs over all primes not exceeding x. In this paper, we give an asymptotic formula for each Sr,m(x) as x → ∞. The case where x is the nth prime number is of particular interest. Here we use a method developed by Salvy to give an asymptotic formula for Sr,m(pn) as n → ∞, which generalizes, for instance, the previously known one for S1,0(pn), the sum of the first n prime numbers.

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(Concerned with sequences A000040 A007504.)

Received December 17 2018; revised versions received December 19 2018; March 29 2019; July 8 2019. Published in Journal of Integer Sequences, August 23 2019.

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